### STRUCTURES - AREA METHOD FOR BEAM DESIGN

The area method for beam design simplifies computation of shear forces and bending moments and is derived, referring to the following diagrams:
2 Beam diagram
3 Shear diagram
4 Bending diagram

The area method may be stated:

•  The shear at any point n is equal to the shear at point m plus  the area of the load diagram between m and n.

•  The bending moment at any point n is equal to the moment at point m plus the area of the shear diagram between m and n.

The shear force is derived using vertical equilibrium:

∑ V = 0;   Vm - w x - Vn = 0;  solving for Vn

Vn = Vm-wx

where w x is the load area between m and n (downward load w = negative).

The bending moment is derived using moment equilibrium:

∑ M = 0;    Mm + Vmx – w x x/2 - Mn = 0;  solving for Mn

where Vmx – wx^2/2 is the shear area between m and n, namely, the rectangle    Vm x less the triangle w x^2/2.  This relationship may also be stated as Mn = Mm + Vx, where V is the average shear between m and n.

By the area method moments are usually equal to the area of one or more rectangles and/or triangles.  It is best to first compute and draw the shear diagram and then compute the moments as the area of the shear diagram.

From the diagrams and derivation we may conclude:

•  Positive shear implies increasing bending moment.
•  Zero shear (change from + to -) implies peak bending moment (useful to locate maximum bending moment).
•  Negative shear implies decreasing bending moment. Even though the forgoing is for uniform load, it applies to concentrated load and non-uniform load as well.  The derivation for such loads is similar.